Vladimir,
Hilbert spaces can only cope with number systems that are division rings. Only three suitable division rings exist. They are the real numbers, the complex numbers, and the quaternions. These numbers are used to specify inner products and eigenvalues of operators. Octonions and bi-quaternions are not division rings. Quantum physicists apply Hilbert spaces to model their theories.
Elementary particles are elementary modules and pointlike objects. At every instant, they take a precise spatial location. Thus they hop around in a stochastic hopping path. The hop landing locations form a coherent swarm. That swarm owns a location density distribution. The hop landing locations are generated by a stochastic process that owns a characteristic function, which is the Fourier transform of the location density distribution and acts like a displacement generator. The characteristic function and the location density distribution lead to Heisenberg's uncertainty relation. Elementary modules configure higher level modules. There, similar relations between swarms and characteristic functions exist.